 mohanty59supriya

2022-07-16

Prove that if x is a rational number and y is an irrational number,then x+y is an irrational number. If in addition, x≠0, then show that xy is an irrational number. Jeffrey Jordon

Hint: Consider $\left(x+y\right)-x$

Explanation:

As is very often the case, we do not need to write this as a proof by contradiction. We can prove the contrapositive directly.

We can prove directly:

x is rational $⇒$ (x+y is rational $⇒$ y is rational)

(using $a,b\in Q⇒a-b\in Q$ -- that is, Q is closed under subtraction)

Therefore (by contraposition of the imbedded conditional)

x is rational $⇒$ (y is not rational $⇒$ x+y is not rational)

This is logically equivalent to

(x is rational & y is not rational) $⇒$ x+y is not rational)

Suppose p and $¬q$ and r .

Prove a contradiction and conclude that if p and $¬q$, then $¬r$

By contrapositive

Suppose p and $¬r$. Prove that q.

Conclude that If p and $¬q$, then r.

The two methods are very closely related and I don't know of anyone who accepts one and not the other. (Although many/most/all intuitionists refuse to accept either contradiction or contrapositive.)

Proof by contrapositive

Suppose that x is rational and $x+y$ is rational.

Then the difference $\left(x+y\right)-x=y$ is rational.

Hence is we know that y is irrational, then $x+y$ must have been irrational.

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